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Breccia, Adriana (2004) Formal bankruptcy: strategic debt service with

senior and junior creditors. Working Paper. Birkbeck, University of London,

London, UK.

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ISSN 1745-8587

Birkbeck Workin

g

Pa

p

ers in Economics & Finance

School of Economics, Mathematics and Statistics

BWPEF 0411

Formal Bankruptcy: Strategic Debt

Service with Senior and Junior

Creditors

Adriana Breccia

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Formal Bankruptcy: Strategic Debt Service with

Senior and Junior Creditors

Adriana Breccia

Economics Working Papers Series,

Birkbeck August 2004.

Abstract

A crucial aspect of debt restructuring is the redistribution of value among many

diverse interests, differing in priority, collateral and bargaining power. Focusing

on renegotiable debt contracts in a continuous-time framework, we characterise the

U.S corporate bankruptcy renegotiation (Chapter 11) in a game-theoretic

frame-work. In formal bankruptcy, the equity holders can renegotiate with one creditor at

a time. Unlike previous studies, this allows us to endogenise not only the bankruptcy

threshold but also the impairment strategy. Moreover, our game-theoretic setting

allows us to explicitly introduce and accommodate varying bargaining powers of

claimants. First, we show that there exists a unique sequence of equilibrium

re-structuring plans and impairment strategies which allows us to derive simple and

intuitive closed-form solutions for pricing different classes of debt. Secondly, the

model provides a theoretical explanation for cases of seniority reversal. Thirdly, we

derive sufficient conditions for the senior credit spread to be smaller than the junior

one at all levels of the state variable.

Comments are welcome. We are very grateful to Giovanni Barone-Adesi, John Brierley, William

Perraudin and Ron Smith for helpful suggestions and discussions. School of Economics, Mathemat-ics and StatistMathemat-ics, Birkbeck College, University of London, Malet Street, London, UK; e-mails:

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Introduction

This paper examines the role of priority on corporate renegotiable debt under the U.S.

bankruptcy code (Chapter 11). Unlike previous literature on renegotiable debt, we

en-dogenise not only the bankruptcy threshold, but also the order in which classes of debt

(with different level of priority) are restructured. In fact, according to Chapter 11, some

classes of debtholders can be excluded from the renegotiating table as long as they receive

the promised contractual payments, that is, such classes are left “unimpaired”.

Interest-ingly, we recognise that this aspect of Chapter 11, well known to legal scholars, largely

enhances the set of strategic actions available to the equity holders. The equity holders

must choose the threshold level to default as well as the class/classes of creditors to

im-pair. Therefore, in this paper, bankruptcy threshold and impairment strategy are jointly

and endogenously determined.

In contrast to previous studies on strategic default, introducing impairment strategies

allows us to capture absolute priority reversals not just amongst debt and equity, but also

among different layers of debt. In fact, we show that what makes the actual debt priority

structure is the optimal impairment strategy, which is uniquely defined by the value of

collateral, bargaining power and face value of each class of debt. This result is in line

with Rajan and Winton (1995), who argue that collateral makes the effective priority of

debt1.

Most important, we show that absolute priority violation amongst senior and junior

debtholders can lead to reversal in the credit spreads. Moreover, for spread reversals not

to occur we provide a simple and intuitive sufficient condition expressed in terms of face

values, collateral and bargaining powers of creditors. Therefore the strategic impairment

characterised in our setting is of great relevance when pricing different classes of debt.

Focusing on renegotiable debt contracts, within a continuous-time pricing framework,

we formalise the bankruptcy system as a judicially supervised bargaining process between

the claimants and the equity holders. Our game-theoretic framework can be compared

with Brown (1989), firstly in that our set of bankruptcy rules captures the main features

of Chapter 11. Secondly, we both stress the importance of those rules which give the

equity holders the option to renegotiate with one creditor at a time 2 in a sort of ‘private

renegotiation’. Briefly, a Chapter 11 filing triggers two essential rules: i) the equity

1In particular, they focus on bank loans, typically highly secured and containing seniority clauses. 2Such as the equity holders’ exclusivity period and the impairment rules allocating veto power in

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holders have the exclusive right to file a first restructuring plan3 and ii) those claimholders

left unimpaired by a plan4 lose their veto power. Therefore, in formal bankruptcy, the

equity holders have the option to renegotiate with one creditor at a time in a sort of

‘private renegotiation’ while excluding unimpaired creditors from the negotiating table.

This possibility is widely exploited in reality, and it largely enhances the equity holders’

strategic behaviour in bankruptcy. It is actually rare the case where a debtor files a

Chapter 11 plan which impairs all claims at once. Quite commonly, a restructuring plan

leaves a certain number of classes unimpaired (most typically, the secured senior classes)

and impairs all or some unsecured classes. In Table A, we report the classification of

claims, the impairment of classes and the solicitation of votes relative to six companies who

proposed Chapter 11 restructuring plans which have all been confirmed by the Bankruptcy

Court5. In all six cases, there is always one or more classes of claims which are left

unimpaired by the plan6, and therefore, as ruled in the Confirmation Orders, those classes

become “non-voting” classes and they are deemed to accept the plan. However, it is

feasible to have a plan confirmed even if the impairment structure does not match the

priority structure. That is, senior classes are impaired while non-senior classes or junior

classes are left unimpaired7.

We capture the ‘impairment strategy’ by modelling the first stage of the restructuring

as a ‘private renegotiation’ where the first equity holders plan consists of a take-or-leave-it

offer to the impaired class/classes -hence, the ‘voting’ classes.

Also, according to Chapter 11 rules, on rejection of the first equity holders’ plan, any

player is allowed to file for competing plans. The bankruptcy code does not set any specific

agenda rules concerning subsequent proposals. Many different ways of modelling the

second stage of the renegotiation are possible8. We use an axiomatic approach and assume

that a three-player Nash bargaining outcome is achieved in the case of a disagreement

3The plan goes ahead if approved by all claimants.

4A creditor is said to be unimpaired if the plan calls for no scaling down of the coupon payment

scheduled in the existing contract.

5Data in Table A are taken from the ‘Plan Disclosure’ and ‘Confirmation Order’ from the U.S.

Bankruptcy Courts of: Southern District of New York, District of Delaware and District of Maryland.

6Of course, this is without taking into account “non-financial classes”, that is classes such as Priority

Claims (unpaid wages), Tax-Claims or Administrative Claims.

7For instance, in Table A, Vero Electronics plan leaves class 3 -‘Other Secured Claims’- unimpaired

and impairs class 2 -‘Senior Secured Credit Facilities’. Cases like this are not the most common, but not

even rare.

8For instance, Brown (1989) assumes that the Bankruptcy Court determines the order in which

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in the ‘private renegotiation’. In contrast to Brown (1989), we recognise the limit of a

‘refereed’ bargaining system. This ideally enables claimants to bring their own rights to

a court which cannot be influenced by parties and which automatically enforces rules.

Nevertheless, as argued by legal scholars, Chapter 11’s rules are idiosyncratic and judicial

discretion is granted in many circumstances9. Therefore, more realistically, our bargaining

setting account for exogenous asymmetries between parties which might reflect the ability

of claimholders to influence the court and extract a better package of concession during

renegotiation. In line with Welch (1997), such asymmetry can be explained as a reflection

of different organisation skills and reputation benefits10. Even though we do not attempt

to model the source of asymmetry, we introduce an exogenous bias in the Nash game which

may encompass a wide range of realistic circumstances whilst keeping the game-theoretic

framework simple and flexible. Therefore the importance of claimholders’ heterogeneity in

terms of bargaining power is stressed through an asymmetric Nash axiomatic bargaining

which jointly involves equity holders and creditors (a senior under-secured and a junior

creditor).

A number of papers have incorporated strategic interactions between debt and equity

holders during renegotiation. In comparison to the literature focusing on strategic debt

service, the distinctive feature of our model is the expansion of the set of strategic actions

available to the equity holders. As mentioned, the formal reorganisation accomplished

under Chapter 11, allows the equity holders to renegotiate vis a vis one creditor at a

time. Therefore, the strategic decision concerns not only the timing of bankruptcy but

also the creditor type, that is whether defaulting on the junior debt, the senior debt or

both. Moreover, the set of strategic actions is additionally expanded by recognising the

possibility of filing for formal renegotiation a number of times. Therefore, our setting

allows for sequential restructuring of different classes of debt.

Our results are as follows.

First, we show that there exists a unique sequence of equilibrium plans which,

de-pending on the parameterization, either impairs the junior creditor first and the senior

one later or vive versa. The equity holders’ ‘impairment strategy’ univocally depends on

factors such as the face value of claims, the bargaining power of players and the value

of collateral. Furthermore, the equilibrium impairment strategy derived is simple and

intuitive.

9See Kordana and Posner (1999) for a critical and exhaustive analysis of Chapter 11’s rules.

10As argued by Welch, unlike bond-holders a bank can have a good reputation effect for “tough

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Second, our valuation of the senior (under-secured) and junior debt, through simple

closed-form solutions, provides some relevant results in terms of credit spreads.

Interest-ingly, the senior credit spread is not necessarily smaller than the junior one at all levels of

the state variable. This could not be the case in a pure liquidation scenario `a la Merton, or under a restructuring system such as the UK Insolvency Law where equity holders have

less control over the renegotiation process11. The privilege of being senior creditor applies

in liquidation where, under absolute priority rule (APR), the line of priority and the

col-lateral are the fundamental and sole factors in determining the claim value. However, in

renegotiation players split the surplus generated by escaping inefficient liquidation,

there-fore the overall package of concession extracted also depends on the bargaining power of

creditors and the timing in which claims are restructured, that is the impairment strategy.

Furthermore, we provide a sufficient condition for avoiding spread inversions. Spread

inversions does not occur if the junior creditor is restructured/impaired before the senior

creditor12. In turn, such impairment strategy results to be the equilibrium strategy if

the senior claimholder has: (i) sufficiently low unsecured face value relatively to the

junior13 and/or (ii) high liquidation value 14 and/or (iii) sufficiently bargaining power

relatively to the junior creditor. Also, it is interesting to note that, in the case where the

senior and junior creditors are both unsecured, the priority of claims does not affect the

impairment strategy and, hence, the possibility of spread reversals. What matters is only

the bargaining power and face values of creditors (that is, point (i) and (iii) above). In

particular, the spread is higher for the creditor with higher ratio of face value to bargaining

power and this occurs regardless of the priority structure.

11The UK Insolvency Law has generally been perceived to allocate substantial privileges to creditors,

particularly to senior creditors holding a floating charge – that is, a floating collateral attached to the overall company’s assets over which the company retains management autonomy. Yet, recently, new insolvency provisions contained in the 2002 UK Enterprise Act seem to reduce the power of senior

creditors in the event of default by circumscribing the administrative receivership. The administrative receiver, appointed by the holder of a floating charge, had the primary duty to realise sufficient assets to pay creditors with higher priority. As argued by Gower (1992), in very few cases the receiver manages

so skilfully that the company is restored to solvency. This might be one of the reasons to move the law toward Chapter 11 style – less creditor friendly.

12The senior claim will be restructured only if the firm value continues deteriorating and falls below a

certain level which triggers a new bankruptcy proceeding.

13That is, the unsecured portion of the senior face value is sufficiently small relative to the junior face

value.

14As in Mella-Barral Perraudin (1997), in our strategic default equilibrium the senior claim value in

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Third, our in-court restructuring can provide a benchmark for out-of-court

restruc-turing of single claims. We assume that there are no renegotiation costs in the formal

bankruptcy proceeding. Therefore, extending this assumption to out-of-court

renegotia-tion implies that our equilibrium in the formal renegotiarenegotia-tion also provides the equilibrium

in private workouts. In this perspective, our setting justifies the possibility of out-of-court

debt forgiveness and strategic default on a single class of claims. This result might be

rel-evant when recognising that, although debt issues often contain cross default provisions,

this does not generally prevent private workout and strategic default on a single creditor.

Indeed, our equilibrium is consistent with the possibility of out-of-court strategic default

on single class of claims accommodated by other classes of claimants.

Fourth, we show that given the overall level of leverage the value of the firm does not

depend on the allocation of face value amongst senior and junior creditors because

strate-gic debt service eliminates direct bankruptcy costs. Therefore, our result of irrelevance

of debt priority structure is consistent with the Modigliani-Miller theorem. In addition,

this result extends and refines Mella-Barral, Perraudin’s results (where bankruptcy costs

from inefficient liquidation are eliminated through strategic bankruptcy) to a scenario

with multiple creditors.

The remainder of the paper is organised as follows. After a brief review of the related

literature, in Section 1, we introduce our main assumptions about the value of the firm

when it continues to operate and when it is liquidated. In Section 2, the bankruptcy rules

defining the game-theoretic structure of the formal renegotiation are set out. In Section

3, we show a simpler version of the model, describing a formal renegotiation between the

equity holders and a single creditor. This simplified scenario allows the reader to easily

understand the mathematical methodology extensively used in the next sections. This

section can also be understood as an application or an extension of MBP technique within

our bargaining settings. In Section 4, we present the model with two classes of debt. In

Section 5, we summarise and explain our results, in terms of timing of bankruptcy (and

impairment strategy), claim values, and restructuring plan. In Section 6, we consider the

implications of our previous results on the credit spreads. Section 7 concludes.

Related Literature Overview

Initiated by Merton (1974), the traditional financial option approach to debt valuation

does not include the possibility of debt restructuring and has the shortcoming of

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is that financial distress is often accompanied by renegotiations, debt rescheduling,

for-giveness, and rarely by liquidations15.

The new approach to value corporate debt incorporates strategic considerations and,

to a more limited extent, bargaining issues. This includes the work of Leland (1994),

Leland and Toft (1996), Anderson and Sundaresan (1996) and Mella-Barral and Peraudin

(1997). Leland (1994) endogenises the restructuring threshold by allowing the payment

of promised coupons through additional equity issues until the equity value is driven to

zero. The initial paper by Leland (1994) has been refined and extended in several

direc-tions, including finite maturities and cash payout, by Leland and Toft (1996). Strategic

bankruptcy has been extensively refined by Mella-Barral and Perraudin (1997) who, in

coninuous-time setting, model strategic debt service when bankruptcy is costly16.

Repre-senting two polar cases, where either the equity holders or the creditor can make

take-or-leave-it offer, they conclude that strategic debt service results in deviations from absolute

priority intuitively because bankruptcy costs from inefficient liquidation allow equity

hold-ers to extract concessions from bond-holdhold-ers. Moreover, their result significantly increases

corporate credit spreads for reasonable parameterization of the model. Exploring two

al-ternative bargaining formulations, such as strategic debt service and debt/equity swap,

Fan and Sundaresan (2000) endogenise dividend policy and the optimal value of the firm

under the two alternate renegotiations.

Even though to a limited extent, most of the recent works, including papers by

Ander-son and Sundaresan (1996), Mella-Barral and Perraudin (1997), Hege and Mella-Barral

(2000), Hege and Mella-Barral (2002) and Fan and Sundaresan (2000), attempt to

ad-dress the importance of claimholders’ bargaining power. This limited literature recognises

the possibility that players might be heterogeneous in terms of bargaining power and, in

turn, the value of renegotiable claims might reflect such a heterogeneity. Yet, apart from

Fan and Sundaresan (2000) who accommodate varying bargaining powers, polar cases are

characterised regarding the distribution of bargaining power amongst equity holders and

creditors. To a certain extent, our paper can be compared to Fan and Sundaresan (2000)

in that, similarly, they characterise a Nash axiomatic solution and explicitly account for

varying bargaining power. Nevertheless, in contrast to Fan and Sundaresan (2000), we

stress the strategic interaction between three diverse players who can renegotiate the debt

service by mean of temporary concessions (strategic debt service, as defined by Anderson

15As reported by Franks and Torous (1989), after the introduction of the 1978 U.S. Bankruptcy Act,

the number of firms seeking for bankruptcy protection has tremendously increased.

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and Sundaresan (1996) and MBP (1997)), while Fan and Sundaresan (2000) focus on

two formulations of renegotiation (such as strategic debt service and debt/equity swap)

between two players.

Looking at the renegotiation framework, those few papers on strategic bankruptcy

accounting for bargaining power considerations, in particular Mella-Barral and Perraudin

(1997), Hege and Mella-Barral (2000), Hege and Mella-Barral (2002) and Fan and

Sun-daresan (2000), incorporate renegotiation by modelling private workouts where generally

disagreement triggers liquidation together with the Absolute Priority Rule. Corporate

debt valuation models of out-of-court renegotiations provide a useful framework to

anal-yse private workouts. However, we stress that a formal bankruptcy proceeding, such as

Chapter 11, differs from private workouts in a number of dimensions which affect the

values of corporate securities. Hence the applicability of informal renegotiation models to

in-court renegotiation system may be limited.

As argued by Sundaresan (2000), “the existence or absence of a bankruptcy code

and its perceived friendliness to borrowers or lenders, is a matter of significance in these

markets. Yet, we have very few pricing theories that have explicitly addressed this as a

structural issue in the determination of spreads”. The key role played by the bankruptcy

code in the valuation of renegotiable debt is yet to be modelled satisfactorily (Sundaresan

(2000)).

Partly, the lack of structural issues can be attributed to the fact that the concern in

pricing corporate debt has been more focused on liquidation than renegotiation scenarios,

and liquidation rules can be simply accounted for by incorporating the Absolute Priority

Rule17.

Furthermore, it is widely recognised that Chapter 11’s rules are quite tedious and

modelling the formal renegotiation might add excessive complications without any further

insight beyond the trivial fact that ‘strong’ players may obtain a bigger share of the ‘pie’

during renegotiation. The recent piece of literature on strategic bankruptcy and strategic

debt service seems to share this perspective and, to some extent, we share it as well, at

least in the situation where a firm has issued a single class of claims.

17See for instance Merton (1974), Black and Cox (1976), Longstaff and Schwartz (1995) where the

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1

Firm Value: Basic Assumptions

The value of the firm is driven by an underlying cash flow process, pt, which follows a

geometric Brownian motion with drift µ and volatility σ. For simplicity, we assume that there are no variable costs and the scrapping value of the firm, γ, is constant. Moreover agents are risk neutral and fully informed, with a risk-free interest rate r.

As the business can be shut down, the value of the firm can be written as

V(pt, p) =

pt

r−µ +

γ − p

r−µ

pt

p

λ

where λ is the negative root of the quadratic equation18

r−µλ− σ

2

2 λ(λ−1) = 0 and is equal to

λ= −(µ−σ

2/2)p

(µ−σ2/2)2+ 2σ2r

σ2 .

The value of the firm is maximised when the threshold level for shutting down, p, is such that

p= arg maxV(pt, p),

which results into

p= λ

λ−1γ(r−µ).

In order to capture the idea that liquidation is generally inefficient as direct bankruptcy

costs arise, we assume that at any time the firm can be liquidated and sold to a potential

new owner who is as efficient as the original owner19. The liquidation sale of the firm

occurs according to a Nash bargaining situation between the initial owner (i.e. the equity

holders) and the new owner, where the terms of the agreement define the firm sale price,

say VL(pt), which is therefore the unknown of the bargaining problem. As well known,

the axiomatic solution to the Nash bargaining problem can be found by maximising the

“Nash product”, say N P, which is the product of the differences between the agreement and disagreement payoff for each player. In our case, the agreements payoffs are: VL(pt)

to the initial owner (that is, the sale price agreed on and received by initial owner) and

18We do not provide here the derivation of the firm value through the stochastic calculus. We refer the

reader to Dixit and Pindyck (1994) for a general analysis of entry and exit decision under uncertainty. See also MBP (1997), for the similarity to our analysis.

19This means that the new owner can generate a valueV(p

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V(pt)−VL(pt) to the new owner20. The disagreement payoffs are simply: γ to the initial

owner (i.e. if no agreement is reached, the firm is sold piecemeal at its scrapping value)

and zero to the new owner. Also, we do not restrict this bargaining situation to be

symmetric and therefore one can imagine that the initial owner has a relative bargaining

power α ∈ [0,1) and, hence, the new owner has bargaining power 1−α. Given these specifications, the Nash product can be written as

N P = (VL(pt)−γ)α(V(pt)−VL(pt))1−α

and it can be easily found that the agreed sale price which maximises the Nash product

is

arg maxN P(VL(pt)) = VL(pt) = αV(pt) + (1−α)γ. (1)

Therefore from our bargaining formulation, the liquidation value is a weighted average

betweenV(pt) andγ, and hence it is always greater or equal to the scrapping value. Even

though this specification of liquidation value, VL, might partly resemble the formulation

by Leland (1994) and Leland and Toft21 (1996), our formulation of V

L is more in line

with Mella-Barral, Perraudin (1997) in that their liquidation value is always below the

maximum firm valueV(pt) but never below the scrapping value of the firm. Moreover their

bankruptcy costs become zero as the state variable, pt, approaches the optimal shutting

down trigger, which is also confirmed by our bargaining formulation of22 V L.

Furthermore, we assume that the firm has issued a perpetual debt with face value F. The debt is allocated over two classes of claims, a senior and and a junior claim with

face values respectively Fs = bs/r and Fj =bj/r (and Fs+Fj =F) where bs and bj are

the contractual coupon payments. To make the problem interesting we assume that the

senior claim is under-secured, i.e.23 F

s> γ.

20The new owner receives the firm value (free of debt) which isV(p

t) and pays the agreed priceVL(pt) to the initial owner.

21Their formulation isV

L=αV (in our notation).

22Note, the bankruptcy costs correspond to the difference V(p

t)−VL(pt) which also rewrites as (1−

α)(V(pt)−γ). Hence the bankruptcy costs converge to zero aspt tends to the optimal shutting down trigger p.

23If the senior claim is fully secured renegotiation involves only the unsecured or under-secured claim.

Even though we do not analyse this case, we show in Section 3 how renegotiation resolves when the firm has issued a single class of under-secured claims. Extending this scenario by adding up a fully secured

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2

Bankruptcy Rules

This section shortly describes the basic assumptions about the formal bargaining

frame-work. The aim is to provide a set of rules to stylise the renegotiation process in the event

of default. Our settings are consistent with the U.S. corporate bankruptcy regulation

(Chapter 11).

Timing of Bankruptcy. As in Mella-Barral, Perraudin we assume that the firm can

voluntarily go into bankruptcy by ceasing paying the contractual coupon. Strategic

bankruptcy implies that the equity holders are free to issue new equity to cover operating

losses. As soon as the firm stops meeting its contractual obligation a formal renegotiation

procedure is triggered.

First Proposal and Impairment Rule. Following the US bankruptcy code for

cor-porate renegotiation (Chapter 11), we assume that the equity holders have an exclusive

right to propose a first reorganisation plan24, which must be approved by all claimants25.

Moreover, again as in Chapter 11, we assume that a creditor cannot reject a plan if

he/she receives cash equal to the face value of his claim or the plan calls for no scaling

down of the coupon payment scheduled in the existing contract. In this case the creditor

is said to be unimpaired26 by the plan and loses his veto power27. This ‘impairment’ rule

is useful to equity holders who can thus negotiate with one creditor at a time.

To sum up, at this first stage of the renegotiation, the equity holders have the right

to make a take-it or leave-it offer, which impairs one or both creditors. If this offer is

accepted by the impaired creditors (as the unimpaired lose veto power) the game ends

and restructuring goes ahead. This initial stage is referred to as ‘private game’.

Subsequent Proposals. On rejection of the first proposal, without any time delay,

the renegotiation moves to a second stage in which any player is allowed to file competing

plans.

The bankruptcy code does not set any specific agenda rules concerning subsequent

proposals. Therefore, on rejection of the equity holders’ proposal, the ‘rules’ of the game

24Bankruptcy Reform Act 1978, Section 1121.

25In Chapter 11, more precisely, approval of two-thirds majority within each class is required but to

keep the bargaining simple we treat each class of claimants as a single agent.

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become in a way unbiased towards different players. The result of the renegotiation

therefore can only depend on the ability of players to propose reorganisation plans. Many

different ways of modelling the second stage of the renegotiation are possible28. We opt

for a simple framework designed to highlight the effect of the different bargaining power

of players who might well differ in organisational skills. The formal structure of our

framework does not favour any player particularly.

We assume, at this stage, that the outcome of the renegotiation is provided by a Nash

axiomatic solution where players split the gains generated from restructuring the business

according to their bargaining strength29. The generalised Nash axiomatic approach seems

to best fit a formal renegotiation which implies the existence of a referee and requires

players to commit to their disagreement payoffs. Moreover the Nash axiomatic approach

is easy to handle and extends unchanged to multilateral bargaining situations. It also

provides a general and flexible result by allowing for asymmetric bargaining power30.

If no agreement is reached at this stage, without time delay, the firm goes into

liq-uidation, therefore the disagreement payoffs in the Nash bargaining correspond to the

liquidation payoff of each claimant.

More formally, let i = e, s, j denote the equity holders, the senior and the junior creditor respectively. As well known31, in the Nash axiomatic bargaining player ‘i’ can

guarantee his/her disagreement payoff (i.e. liquidation payoff), Li, plus a share of the

surplus, V(pt)−VL(pt), which depends on the relative bargaining power32. Therefore, in

28For instance, Brown (1989) assumes that the Bankruptcy Court determines the order in which

pro-posals are voted and each proposal might rank first, second or third in the agenda with same probability.

29Introducing an exogenous bargaining power simply captures the idea that players might influence the

Bankruptcy Court as Chapter 11 is a judicially supervised bargaining process (see A. Schwartz, 2002).

Moreover in a mandatory Bankruptcy system where parties must use the state supplied procedure it is more realistic to imagine that courts are not perfectly informed, and therefore the judge’s decision is conditional to the information disclosed by the parties. Therefore the bargaining skills of players at

influencing the Court’s decision are crucial (see Welch, 1997).

30Fan and Sundaresan (2000) use the Nash axiomatic approach, with asymmetric bargaining power, to

model strategic debt service (as well as debt/equity swap) in a private workout between equity holders

and a single class of debt holders. Apart from the similarity in the Nash bargaining, their study crucially differs from the current model. We deal with multiple classes of debt in formal bankruptcy while they compare strategic debt service and debt/equity swap when there is only one class of debt holders.

31See Binmore and Dasgupta (1987) for an extensive analysis of the Nash bargaining solution. We

also refer to Fan and Sundaresan (2000), for the similarity to the current paper. For a more detailed

derivation of the Nash bargaining solution when uncertainty is modelled through a geometric Brownian motion, see also Perraudin and Psillaki (1999).

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the Nash bargaining, player ‘i”s claim value is equal to

Pi =ξi(V(pt)−VL(pt)) +Li, (2)

where ξi and Li denote the bargaining power and the liquidation payoff of player ‘i’

re-spectively. We will refer to this stage as the ‘joint game’.

Liquidation. If the firm is liquidated, the Absolute Priority Rule (APR) applies.

For-mally, under APR the liquidation payoffs for s, j and e are respectively

Ls = min{Fs, VL} (3)

Lj = min{Fj, VL−Ls} (4)

Le = max{VL−Ls−Lj,0}. (5)

Terms of Contracts and Enforceability. In practice, the renegotiation can be

im-plemented by reducing coupon payments while the firm is in a default region, through a

variable, state contingent, debt service flow as in MBP (1997). Therefore in what follows

we will focus on restructuring plans which consist of piecewise right-continuous service

flow function of the state variablept, denoted asbi(pt) fori=s, j with bi(pt)< bi.

As the equity holders cannot commit to remain out of bankruptcy and can trigger

renegotiation at any time, creditors will underwrite a state contingent contract only if

this is self enforceable on the equity side (in the sense that the equity holders will not

have incentive to deviate). It is reasonable then to imagine that players bargain over

short term contracts, i.e. instantaneous payoffs, and that they can renegotiate

continu-ously. Though this might be not realistic, a continuous time renegotiation provides the

necessary benchmark to derive a self enforceable contract.

We conclude this section by summarising the bankruptcy rules and introducing some

nota-tion. When the equity holders trigger bankruptcy the renegotiation is structured as a

two-step game without time delay in between the first and the second stage. In the first game,

the “private game”, having the right to the first proposal, the equity holders make a

take-it or leave-take-it offer which might impair the junior credtake-itor, the senior or both credtake-itors. Let

the set of plans proposed by the equity holders beP ={Pe, Pj, Ps :Pe+Pj+Ps =V(pt)}.

At the second stage, “joint game”, each player can guarantee a payoff, denoted as Pi for

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plan will be accepted at the first round by an ‘impaired claimant’, denoted by ‘i’, only

if Pi ≥ Pi. Panel A shows the two-step game in which, at the first round, the equity

holders choose whether to impair: i) the junior creditors by proposing a plan Pj ⊂ P

where Pj = {Pe, Pj, Ps : Pe ≥ Pe, Pj ≥ Pj, Ps < Ps}, ii) the senior creditor by

propos-ing a plan Ps ⊂ P where Ps = {Pe, Pj, Ps : Pe ≥ Pe, Pj < Pj, Ps ≥ Ps} or iii) both

the senior and the junior creditors and in this case the only feasible plan, say Ps,j ⊂ P,

must guarantee that each player receives his reservation payoff in the joint game, i.e.

Ps,j ={Pe, Pj, Ps :Pe= Pe, Pj = Pj, Ps= Ps}.

Panel A: Bankruptcy game

Private game Joint game Liquidation

r r r

‘e’ offers:

Pj impairing ‘j’, orPs impairing ‘s’,

orPs,j impairing both.

on rejection by:

j

s

at least one

player

−→

i=e,j,s, share the

‘surplus’ according

to a payoff Pi.

on rejection

by at least

one player

−→

Liquidation

payoffsLi, for

i=e, j, sand

P

Li =VL

3

A Simplified scenario: one creditor only

Before describing the renegotiation with two creditors, we briefly show a simplified

sce-nario in which the firm has issued only one class of under-secured claims, with face

value Fc = bc/r > γ33. This simplified scenario allows the reader to easily understand

the mathematical methodology we will extensively use in the next section in order to

calculate claims, equity value and restructuring plan (i.e. a debt service flow function

bc(pt)< bc). The technique shown here is similar to that developed by Mella-Barral,

Per-raudin34. Therefore, this section can also be understood as an application or an extension

of Mella-Barral, Perraudin’s technique within our bargaining setting.

33In this section, we use the subscript ‘c’ to denote the debt holder.

34As the purpose of this paper is quite different from that of Mella-Barral, Perraudin, we limit this

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As there is only one creditor, some of the bankruptcy rules defined in Section 2 are

redundant here. More specifically, it is trivial to notice that the equity holders cannot

strategically use the impairment rule because there is no other class of claims which can

be left unimpaired. Moreover, the private game becomes redundant here, in the sense

that the equilibrium shares in the private game guarantee players with their equilibrium

shares in the joint game. In fact, given the equilibrium partition in the joint game, i)

the claim holder accepts the first equity holders’ proposal only if he/she receives at least

the equilibrium share of the joint game and ii) the equity holders will offer at most the

creditor’s equilibrium share of the joint game. Because there is no time delay between the

private and the joint game, and hence the value of the ‘pie’ does not change, it follows

that the equilibrium shares in the private and the joint game are the same. This allows us

to skip the private game and solve the bankruptcy game by simply solving the two-player

Nash axiomatic bargaining.

Apart from these changes, the model develops within the same setting and

assump-tions introduced in the previous section.

Hypotheses 1A. Assume that there exists a strategy in terms of stopping times, such

that i) when the state variablept crosses a certain trigger, saypc, the equity holders

rene-gotiate with the debt holder in a Nash bargaining (through a service flowbc(pt)< bc).

Therefore, given that pc exists, one can formalise the equity holders’ decision problem

as follows

max

pc

E(pt, pc) (6)

st. E(pt, pc) = V(pt)−C(pt, pc))

C < Fc,

where C(pt, pc) is the debt value for pt≥pc and Fc is the face value of the debt.

Hypotheses 2A.When the state variable is in the range [p, pc] we assume thatVL< Fc.

Because APR applies, this implies thatLe = 0 andLc =VL.

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shares to the equity holders and the creditor become respectively

Pe = ξe(V −VL) (7)

Pc = ξc(V −VL) +VL (8)

with

(

ξi ∈I = [0,1]

P

iξi = 1

for i=e, c.

As the renegotiated shares are already defined by 7 and 8, left to determine are the

trigger pc and the service flow bc(pt).

In order to derive the bankruptcy trigger pc, one can solve the equity holders’

max-imisation problem, or equivalently, the minmax-imisation problem:

min

pc

C(pt, pc) (9)

st. C < Fc

with n C(pt) = Pc for pt=pc

where the debt value, C(pt, pc) can be written as

C(pt) =

bc

r +

Pc(pc)−

bc

r

pt

pc

λ

. (10)

Solving the above problem yields the optimal trigger

p∗c = λ

λ−1

bc/r−γ(1−αξc)

αξc

(r−µ), (11)

with

αξc =ξc(1−α) +α. (12)

Moreover, as shown in Appendix 1, the triggerp∗c guarantees that Hypotheses 2A holds35.

What is left to derive is the restructuring plan. Similarly as in MBP (1997), one can

derive the service flow functionbc(pt) as follows. We know that, under risk neutrality, the

claim value,C(pt) is free of arbitrage opportunity if and only ifbc(pt) solves the differential

equation

rC(pt) = s(pt) +

d

d4Et(Ct+4)|4=0, (13)

35In particular, we prove in Appendix 1 thatF

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where

sc(pt) =

(

bc if pt ∈[p∗c,∞)

bc(pt) if pt ∈[p, p∗c)

subject to the following conditions. First, C(pt) must be continuous in level and first

derivative, which implies

C(p∗c) = Pc(p∗c) (14)

∂C(pt)

∂pt

|p∗

c =

∂Pc(pt)

∂pt

|p∗

c . (15)

This conditions are also known as smooth-pasting conditions. Secondly, equation 13 must

satisfy no arbitrage and no bubbles conditions which correspond respectively to

Pc(p) = γ, (16)

lim

pt→∞

C(pt) =

bc

r. (17)

One can find that the debt value,C(pt), defined by 10, satisfies all the above conditions36.

As shown in Appendix 3, there is a unique function, s(pt), which solves 13, and this

is given by

sc(pt) =

(

bc if pt∈[p∗c,∞)

bc(pt) = αξcpt+ (1−αξc)rγ if pt∈[p, p

∗ c)

with αξc defined in 12.

In Figure 1, we show the renegotiated value of the debt under strategic debt service.

For pt ≤ p∗c, the debt value is always between V(pt) and the liquidation value VL(pt).

Moreover it is equal to either V(pt) or VL(pt) only in the extreme scenarios where either

the creditor or the equity holders have full bargaining power. In these extreme cases our

result is similar to MBP37 and, to a certain extent, this section provides a generalisation

of their results which accommodate varying bargaining powers.

36In fact 10 satisfies conditions 14, 16 and 17 by construction. In Appendix 2, we show that the first

order condition to the equity holders’ maximisation problem, that is∂C(pt, pc)/∂pc= 0, is equivalent to the smooth pasting condition 15 which is, therefore, satisfied.

37There are marginal differences due to the modelling of direct and indirect bankruptcy costs. In

Mella-Barral, Perraudin, when the debt is not renegotiable, on default, the debt holders take over the firm and run the business less efficiently (in terms of net cash flows) than the former management. This assumption allows one to account also for indirect bankruptcy costs, while in our model indirect bankruptcy costs

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Apart from the methodological purpose of this section, we want to stress that the

formal setting of the bankruptcy process does not play a crucial role when there are only

two players at the negotiating table. In fact, as argued, most of the literature on debt

restructuring does not focus on formal bankruptcy but rather on private workouts when

there is a single class of claims. In line with this literature, we have pointed out that most

of the bankruptcy rules appear to be redundant in this simplified scenario. Moreover, the

advantage of formalising the bankruptcy process seems quite limited in terms of economic

intuition. Quite different is the purpose of the bankruptcy rules when there are multiple

claims. As we show in the next section, all of the formal rules are strategically enforced

by the equity holders.

4

The Model

In this section we show how renegotiation is implemented when the firm has issued two

classes of claims as defined in Section 1. In this scenario, none of the bankruptcy rules

described in Section 2 is redundant. On the contrary, one can notice here how our

bankruptcy framework helps to define a unique equilibrium in terms of renegotiating

strategy and restructuring plan.

Hypotheses 1 Assume that there exists a strategy in terms of stopping times, such

that i) when the state variable pt crosses a certain trigger, say pj, the equity holders

start renegotiating with the junior creditor while pt < pj (by proposing a service flow

bj(pt)< bj) and ii) whenpt crosses the trigger ps they start renegotiating with the senior

creditor (through a service flow bs(pt)< bs).

Now, let max{pj, ps} = p∗ and min{pj, ps} = p∗. Consistently within our bargaining

framework the equity holders strategy can be summarised as follows:

when p∗ < pt≤p∗ propose plan Pi, bi(pt) with

(

i=j if p∗ =pj

i=s otherwise (18)

when p≤pt≤p∗ propose plan Ps,j,

(

bj(pt)

bs(pt)

(19)

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Therefore, the problem of the equity holders can be generalised as

max

pj,ps

E(pt) (21)

st. E(pt) = V(pt)−S(pt, ps, pj)−J(pt, pj, ps)

S < Fs

J < Fj,

where E(pt), S(pt) and J(pt) denotes the values of the equity, the senior and the junior

debt and Fs and Fj are the face values.

Hypotheses 2 We assume that VL < Fs, when the state variable is in the range [p, ps].

As absolute priority applies, this implies that Le = 0, Lj = 0 and Ls =VL, therefore the

shares in the joint game will be

Pe = ξe(V −VL) (22)

Pj = ξj(V −VL) (23)

Ps = ξs(V −VL) +VL (24)

with

(

ξi ∈I = [0,1]

P

iξi = 1

for i=e, j, s.

4.1

Equilibrium offers and trigger strategy

The purpose of this section is to determine the equilibrium values, E, J and S and the optimal trigger strategy {pj, ps}. We already know, by 22-24, the values of equity, senior

and junior claims when38 p≤pt ≤p∗, therefore the problem is to determine E, J and S

when p∗ < pt ≤p∗.

Our argument to solve the renegotiation runs as follows.

We determine E, J, S and the trigger {pj, ps} under the two possible strategies: CASE

A) ps < pj, i.e. the equity holders impair the junior first, CASE B) ps ≥ pj, the senior

creditor is impaired first (if ps > pj) or jointly with the junior creditor (forps =pj).

In order to determine the equilibrium values in both cases, A and B, we use a no

arbitrage argument (in the remainder point (i)) and a backward solution when solving for

38In fact, if p

∗ = min{pj, ps} = ps, then by Hypotheses 2 VL(ps) < Fs and therefore, for pt < ps, players’ payoffs are given by 22-24. If p∗ = min{pj, ps} = pj, then it is the case that pj < ps and, beingVL(pt) decreasing inpt, it holds from Hypotheses 2 thatVL(pj)< VL(ps)< Fs, therefore again for

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the optimal triggers {pj, ps} (point (ii)).

Then, we show that depending on the level of the parameters, ξi (for i = e, j, s), Fj,

Fs, α, r and γ, a unique equilibrium strategy exists.

CASE A:ps< pj. The equity holders impair the junior creditor first, whenps< pt≤pj,

and then the senior creditor also when p < pt≤ps.

(i) We define here the junior claim equilibrium value under an accepted offer from the

equity holders in the private game.

Proposition 4.1.1 When ps < pt ≤ pj, the smallest offer by the equity holders to the junior creditor is accepted if and only if J(pt, pj) =Pj+Ps−S(pt, ps).

Proof Let us define first an arbitrage strategy. In the private game, whenps < pt ≤pj,

after an equity holders’ offer, the junior creditor can: 1) buy the senior claim atS(pt, ps),

2) reject the equity holders’ plan, therefore losing J(pt, ps) and 3) before the start of the

joint game, sell the senior claim at Ps(pt). The payoff from such a strategy, say Π, is

given by

Π = Pj + Ps−J(pt, pj)−S(pt, ps). (25)

We show in Appendix 4 that Π ≥ 0 under an equity holders’ proposal belonging to Pj.

Therefore an offer from the equity holders, is accepted by the junior creditor, if and only

if Π≤0, that is J(pt, ps)≥Pj+ Ps−S(pt, ps). Then we conclude that the smallest offer

which is accepted by the junior creditor must be such that Π = 0.

Therefore, by Proposition 4.1.1, ‘j”s equilibrium value (under an accepted offer) is such

that Π = 0, which, by 25, implies

J(pt, pj) = Pj+ Ps−S(pt, ps). (26)

(ii) The triggers ps and pj can be found by working backward. When pt is greater

than ps the senior value is equal to

S(pt, ps) =

bs

r +

Ps(ps)−

bs

r

pt

ps

λ

(23)

which, minimised with respect to ps gives39

p∗s2 = λ

λ−1

bs/r−γ(1−αξs)

αξs

(r−µ), (28)

with

αξs =ξs(1−α) +α. (29)

Then, minimising the junior value with respect to pj, which, by 26 and 28, is equal to

J(pt, pj) =

bj

r +

Pj(pj) + Ps(pj)−S(pj, p∗s2)−

bs

r

pt

pj

λ

,

yields the optimal trigger

p∗j1 = λ

λ−1

(bs+bj)/r−γ(1−αξs,j)

αξs,j

(r−µ), (30)

with

αξs,j = (ξs+ξj)(1−α) +α. (31)

CASE B:ps ≥pj. The equity holders impair the senior creditor first, whenpj < pt≤ps,

and then the junior creditor also when p < pt ≤pj.

(i) We derive here the senior claim equilibrium value under an accepted offer in the

private game when pj < pt ≤ ps. One can use a similar argument as the previous case

(point (i)). In Proposition 4.1.1, replacing the word ‘junior’ with ‘senior’, one can

con-clude that the senior claim equilibrium value (under an accepted offer) must be such

that40 Π = 0, which implies

S(pt, pj) = Pj + Ps−J(pt, pj). (32)

.

39We denote the optimal trigger by using the subscript ‘in’, with i = s, j and n = 1,2, to stress

the order in which claims are impaired. Therefore, for instance, when the senior creditor is the second claimant to be impaired we denote the optimal trigger level as ps2 (while when the senior creditor is

impaired first, the trigger level is denoted asps1). 40As in the previous case, the gain, Π, is equal to P

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(ii) As before, by working backward one can find pj first and then ps. Minimising the

junior value, which, at pt> pj is equal to

J(pt, pj) =

bj

r +

Pj(pj)−

bj

r

pt

pj

λ

, (33)

gives

p∗j2 = λ

λ−1

bj/r+γξj(1−α)

ξj(1−α)

(r−µ). (34)

Then, by minimising the senior value with respect to ps, which (by 32 and 34), when

pt> ps, is equal to

S(pt, ps) =

bs

r +

Pj(ps) + Ps(ps)−J(ps, p∗j2)−

bs

r

pt

ps

λ

,

and solving for the optimal trigger yields

p∗s1 = λ

λ−1

(bs+bj)/r−γ(1−αξs,j)

αξs,j

(r−µ), (35)

where αξs,j defined in equation 31.

Unique Equilibrium strategy

So far we have identified two optimal trigger strategies, in CASE A) a strategy {p∗j1, p∗s2}

if the junior is impaired first and, in CASE B) a strategy {p∗j2, p∗s1} if the senior creditor is impaired first (that is p∗s1 > p∗j2) or jointly (p∗s1 =p∗j2) with the junior creditor.

In the remainder of this section we will show that there is a unique equilibrium

strat-egy. The equilibrium is unique if once the equity holders impair creditori=sorj, leaving creditor i 6= i unimpaired, they will not invert strategy in the future (that is, impairing i6=i and paying the full contractual coupon to creditor i).

First, notice that regardless of the fact that the creditor who is impaired first is the

junior (as in CASE A)) or the senior one (CASE B)) the higher bankruptcy trigger does

not change. In fact,

p∗j1 =p∗s1.

Therefore the equilibrium strategy will only depend on the triggers ps2 and pj2.

Let us denote with ¯pthe trigger p∗j1 =p∗s1. One can find, by some simple algebra, that

p∗s2 <p¯ if and only if bj

r αξs,j

ξj(1−α) > F −γ

p∗j2 ≤p¯ if and only if bj

r αξs,j

ξj(1−α) ≤F −γ.

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Proposition 4.1.2 The optimal strategy is unique and there is no ambiguity in the path

of the renegotiation. The equity holders play the optimal strategy

{p∗j1, p∗s2} iff bj

r αξs,j

ξj(1−α) > F −γ

{p∗s1, p∗j2} iff bj

r αξs,j

ξj(1−α) ≤F −γ.

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Proof Whenp∗s2 <p¯(i.e. bj

r αξs,j

ξj(1−α) > F−γ) is also true thatp

j2 >p¯, butp∗j2 is an optimal

trigger only if p∗j2 ≤ p∗s1 = ¯p. Therefore, we can conclude that p∗j2 does not exist in this case and hence there is a unique strategy left, that is {p∗j1, p∗s2}. Under this strategy the equity holders impair the junior creditor first whenpt= ¯pand senior later, whenpt=p∗s2.

If instead bj

r αξs,j

ξj(1−α) ≤ F −γ then p

j2 ≤ p¯ and p ∗

s2 > p¯. Nevertheless, for p ∗

s2 to be

optimal it must be the case that p∗s2 ≤ p∗j1 = ¯p. Similarly as the previous case, we con-clude thatp∗s2 does not exist and there is only one strategy left corresponding to{p∗j2, p∗s1}. Therefore the equity holders impair the senior creditor first (i.e. whenpt = ¯p) and, when

pt=p∗j2, the junior creditor also.

Last, as shown in Appendix 5, one can notice that under both strategies, {p∗s2, p∗j1} and

{p∗s1, p∗j2}, Hypotheses 2 holds, that is VL(ps) < Fs (with ps equal to either p∗s2 or p ∗ s1).

This guarantees that the senior claims is never renegotiated when the senior creditor could

guarantee the full face value in liquidation.

4.2

Restructuring plan

We have derived in the previous section the equilibrium value of claims and the trigger

strategy which defines the optimal timing for restructuring each class of claims. In this

section, we will determine the debt service flow function which guarantees each creditor

with his/her equilibrium values during renegotiation.

Depending on the set of parameters in our model, we have shown that the equilibrium

trigger strategy is either {p∗s2, p∗j1} (if bj

r αξs,j

ξj(1−α) ≤ F −γ) or {p

j2, p∗s1} (if else) with

cor-responding claims equilibrium values, J(pt) and S(pt), defined in CASE A) and B) (of

Section 4.1) respectively. Therefore, according to the two possible sets of parameters

there are two corresponding restructuring plans defined in the following propositions.

Our derivation of the debt service flow functions follows the same line as the derivation

of bc(pt) in Section 3. Therefore, we only present here our results and refer the reader to

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Proposition 4.2.1 If bj

r αξs,j

ξj(1−α) > F −γ the equilibrium restructuring plans impair the

junior and senior creditor according to service flow functions respectively

bj(pt) = αξs,jpt+ (1−αξs,j)γr−ss(pt) for pt≤p

j1 (38)

ss(pt) =

(

bs for p∗s2 < pt

bs(pt) = αξspt+ (1−αξs)γr for pt≤p

∗ s2

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with αξs,j and αξs defined by 31 and 29 respectively.

Proof See Appendix 6.

Proposition 4.2.2 If bj

r( αξs,j

ξj(1−α))≤F −γ the equilibrium restructuring plans impair the

junior and senior creditor according to service flow functions respectively

bs(pt) = αξs,jpt+ (1−αξs,j)γr−sj(pt) for pt≤p

s1 (40)

sj(pt) =

(

bj for p∗j2 < pt

bj(pt) =ξj(1−α)(pt−γr) for pt≤p∗j2

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Proof See Appendix 7.

5

Comments and Summary of results

In this section we recombine and explicit our results in Section 4. The purpose is to

ex-plain our results with particular emphasis on the effect of bargaining powers, face values

and allocation of priority. We group our findings into three categories : A) timing of

bankruptcy and impairment strategy, B) equity and claims values and C) restructuring

plan.

A) Timing of bankruptcy and impairment strategy. The equity holders trigger

bankruptcy, and file a plan Pi orPs,j,41 according to the following equilibrium strategy:

when p∗i2 < pt ≤p¯ file plan P6=i, with

(

i=s if bj

r αξs,j

ξj(1−α) > F −γ

i=j otherwise (42)

when p≤pt ≤p∗i2 file plan Ps,j, (43)

41We remind the reader that a planP

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where

p∗s2 = λ

λ−1

bs/r−γ(1−αξs)

αξs

(r−µ), (44)

p∗j2 = λ

λ−1

bj/r+γξj(1−α)

ξj(1−α)

(r−µ), (45)

¯

p = p∗j1 =p∗s1 = λ

λ−1

(bs+bj)/r−γ(1−αξs,j)

αξs,j

(r−µ), (46)

and

αξs = ξs(1−α) +α, (47)

αξs,j = (ξs+ξj)(1−α) +α. (48)

First, notice that, given the overall face value of the debt, F, the higher bankruptcy trigger is independent of the priority structure of claims. Bankruptcy is triggered the first

time, as soon as the state variable crosses p∗s1 = p∗j1. In other words, given the overall creditors’ bargaining power and the face value F, the default region, [p,p¯= p∗s1 = p∗j1] is independent of the type of creditor impaired first and is not affected by the allocation

of debt amongst creditors. In order to highlight the irrelevance of the debt priority

structure over the default region, one can compare two firms which differ only for their

debt allocation. Imagine, for instance the limiting case of an identical firm (where the

equity holders have the same bargaining power, ξe) with just one creditor with a claim

of face value F = Fj +Fs and bargaining power ξc = 1−ξe = 1−ξs−ξj. As shown

in Section 3, renegotiation starts when pt crosses λλ−1

F /r−γ(1−αξc)

αξc (r−µ) which coincides with the optimal trigger42 p∗s1 =p∗j1 = ¯p.

Therefore, not surprisingly, what defines the default region is the level – not the

allo-cation – of F and the overall bargaining power of creditors vis a vis the equity holders. The higher F and/or ξj+ξs the earlier bankruptcy occurs43.

If, on the one hand, the debt priority structure (i.e. the allocation of face value amongst

senior and junior creditors) does not alter the default region, on the other hand it crucially

determines the timing and the order in which claims will be impaired during bankruptcy.

In a way, one could say that, given the overall face value, the priority structure of claims

determines the ‘default regions of each single claim’. Using this terminology, depending

on the claims face values, there are two possible default regions for each debt-holder: i)

42Simply replace the notationb

j/r+bs/rinstead ofF and notice thatαξc =αξs,j.

43In fact,p

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[p, p∗j1 =p∗s1] and [p, p∗s2], respectively for the junior and the senior, if bj

r αξs,j

ξj(1−α) > F−γ or

ii) [p, p∗s1 =p∗j1] and [p, p∗j2], if instead bj

r αξs,j

ξj(1−α) ≤F −γ

44.

The intuition behind this result is immediate. We know that the equity holders always

start renegotiating atp∗s1 =p∗j1 and when the state variable crosses this trigger they would benefit from reducing coupon payments to the creditor whose face value is relatively high

with respect to the overall face value F. Therefore, given F, if, for instance, bj/r is high

enough the equity holders would benefit from impairing the junior creditor first.

The former consideration is not the only factor determining the order of the impaired

claims. The benefit from impairing the creditor with higher face value must be weighted

against the ‘strength’ of that creditor, that is, the ability to extract a valuable package

of concessions in renegotiation. This depends on the liquidation value of the firm, the

priority of the claim and the creditor’s bargaining power. The higher αand γ, the bigger the liquidation value, which, in turn, strengthens the bargaining position of the senior

whilst weakens that of the junior45. Therefore, when the liquidation value is sufficiently

high, the equity holders impair the junior creditor, who can extract smaller concessions

than the senior claimant. The argument runs similarly and the same conclusion holds

when the bargaining power of the junior creditor is sufficiently small. If this is the case,

again, the equity holders impair first the junior creditor, that is, the ‘weak’ player. All

these factors are captured by the inequality bj

r αξs,j

ξj(1−α) RF −γ which determines the order in which claims are impaired. In fact, for instance, it is more likely that bj

r αξs,j

ξj(1−α) > F−γ

(and hence the junior is impaired first) for high levels of bj/r, α and γ and low level of

ξj, consistently with our intuitive explanation.

Finally, our argument can be clearly summarised by rearranging the above inequality

as

αξs

Fs−γ R

ξj(1−α)

Fj

.

Here, each side of the inequality measures the ‘intensity of the actual bargaining strength’.

Precisely, the termsαξs and ξj(1−α) can be interpreted as actual bargaining strength of creditors in that these terms account for i) the exogenous bargaining power,ξi, and ii) α

which determines the disagreement/liquidation payoff46(which is an endogenous

compo-44When bj

r αξs,j

ξj(1−α) =F−γ,p

j2 =p∗s1, that is claims are jointly impaired and the two default regions

are the same.

45Because the junior creditor is a residual claimant, he/she purely benefits from receiving a share of

the firm continuation surplus, V −VL, while the senior creditor, due to the priority of his/her claim, guarantees also the full liquidation value.

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nent of the bargaining strength). Moreover, in the above inequality, the actual bargaining

strength is measured in terms of units of unsecured face value (that is, Fs−γ and Fj),

which explains why one can refer to each side of the inequality as to intensity of

bargain-ing strength. By usbargain-ing this terminology, we conclude that the equity holders impair first

the creditor with smaller intensity of actual bargaining strength.

B) Equity and claims values. According to the previous trigger strategies, if the

junior creditor is a ‘weak’ player in comparison to the senior one, the equity holders file

a plan:

Pj, ∀ pt ∈(p∗s2,p¯] Ps,j, ∀ pt ∈[p, p∗s2]

⇒ if bj

r

αξs,j

ξj(1−α)

> F −γ (49)

which guarantees creditors with the following claims’ values:

S(pt) =

 

bs

r + Ps(p∗s2)−brs

pt

p∗ s2

λ

if pt> p∗s2

Ps(pt) if pt ≤p∗s2

(50)

J(pt) =

       bj r +

Pj(¯p) + Ps(¯p)−S(¯p)− bj

r pt

¯ p

λ

if pt>p¯

Pj(pt) + Ps(pt)−S(pt) if p∗s2 < pt≤p¯

Pj(pt) if pt≤p∗s2

(51)

If instead the junior creditor is a ‘strong’ player, the equity holders impair first the

senior claimant by filing a plan:

Ps, ∀ pt ∈(p∗j2,p¯] Ps,j, ∀ pt∈[p, p∗j2]

⇒ if bj

r

αξs,j

ξj(1−α)

≤F −γ (52)

which guarantees creditors with the following claims’ values:

S(pt) =

       bs r +

Pj(¯p) + Ps(¯p)−J(¯p)−bs

r pt ¯ p λ

if pt>p¯

Pj(pt) + Ps(pt)−J(pt) if p∗j2 < pt ≤p¯

Ps(pt) if pt≤p∗j2

(53)

J(pt) =

   bj r +

Pj(p∗j2)− bj

r pt

p∗j2

λ

if pt > p∗j2

Pj(pt) if pt≤p∗j2

(54)

(30)

where

Pe = ξe(V −VL) (55)

Pj = ξj(V −VL) (56)

Ps = ξs(V −VL) +VL. (57)

We show in Figure 2, 3 and 4 the equilibrium claim values resulting from three

alter-native scenarios where the junior is impaired first (that is, p∗s2 < p∗j1 = ¯p, as in Figure 2), together (i.e. p∗s2 =p∗j1 = ¯p, Figure 3) or after the senior has been impaired (that is,

p∗j2 < p∗s1 = ¯p, see Figure 4).

One can notice that, given the creditors’ bargaining power ξs+ξj and the face value

F, the priority structure of claims does not affect the equity value. In fact, the debt value, S(pt) +J(pt), remains the same regardless of the allocation of face value amongst

creditors.

This result is in line with MBP (1997), in that the equity value is affected by the

overall face value and the bargaining power of the equity holder. Moreover it extends

MBP results to a multiple creditors scenario, in that the allocation of debt amongst

classes is irrelevant.

Furthermore, by preventing inefficient liquidation, strategic debt service eliminates

direct bankruptcy costs. This result refines the Modigliani-Miller theorem in terms of

irrelevance of the debt priority structure.

We conclude that, in a frictionless market, like the one depicted in our model, whether

there exists a debt optimal priority structure, this cannot be derived from the path of the

renegotiation during bankruptcy.

Further analytical results, directly related to the equilibrium claim values, can be

found in terms of credit spreads. We refer the reader to the next section where the

im-plications of the claim values on the spreads are investigated in detail.

C) Restructuring plan. The equity holders file either a restructuring plan Pj or

a plan Pj, depending on the level of our parameters. After impairing either class of

claims they will propose an equilibrium plan Ps,j.

The equilibrium plan Pj consists of a promise to pay the following coupon flows:

(

bj(pt) =αξs,jpt+ (1−αξs,j)γr−bs

bs

⇒ for pt≤p¯ (58)

(31)

When instead the equity holders file a planPs they promise to pay the pair of coupon

flows

(

bj

bs(pt) =αξs,jpt+ (1−αξs,j)γr−bj

⇒ for pt≤p¯ (59)

which impairs the senior creditor without rescheduling payments for the junior claimant.

After impairing only one creditor by filing a plan Pj orPs, the equity holders will file

the equilibrium planPs,j which consists of a pair of debt service flow functions:

(

bj(pt) = ξj(1−α)(pt−rγ)

bs(pt) = αξspt+ (1−αξs)γr

⇒ for pt ≤

(

p∗s2 if Pj has been filed

p∗j2 else (60)

This equilibrium plan impairs both creditors at once.

In Figure 5, 6 and 7, we show the three possible pairs of debt service flow functions

resulting from three different sequences of bankruptcy plans such as {Pj,Ps,j} (in Figure

5),{Ps,j}(in Figure 6, where claims are jointly impaired at the trigger levelp∗s2 =p ∗ j2 = ¯p)

and {Ps,Ps,j}(in Figure 7).

6

Implications on Credit spreads and risk premia

The purpose of this section is that of showing the effect of our renegotiation framework

on the credit spreads of the two classes of debt. It is intuitive that the opportunity of

rescheduling debt, by allocating the continuation surplus of an economically viable firm,

allows creditors to improve their payoffs as a group. The appropriation of a share of this

surplus gives each creditor the incentive to successfully renegotiate his/her claim vis a

vis the equity holders. The opportunity to renegotiate should therefore reduce the credit

spreads simply because it increases the claim values with respect to a pure liquidation

scenario.

In order to isolate the effect of the renegotiation on credit spreads we first define the

spread and then we decompose it into two kinds of premia: a default and a renegotiation

premium.

Because our coupon bonds are perpetuity the spread, say CS, can be measured as

CSi =

bi

Bi

−r with

   

  

i=s, j

Bs =S(pt)

Bj =J(pt),

Figure

FIGURE 2:  Senior and junior debt values when ps<pj
FIGURE 3: Senior and junior debt values when ps=pj
FIGURE 5: Senior and junior debt service when ps<pj
FIGURE 7: Senior and junior debt service when ps>pj
+3

References

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